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Cell[CellGroupData[{
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Cell[TextData[{
  "This Mathematica notebook is from the WWW project A Visual Dictionary of \
Special Plane Curves. Copyright \[Copyright] 1995-2000 by Xah Lee. (",
  ButtonBox["http://xahlee.org/",
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      URL[ "http://xahlee.org/"], None},
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  ")"
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Cell["Proof of different formulas for astroid", "Section"],

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Cell["Problem", "Subsection"],

Cell["\<\
Given the parametric formula
x=Cos[t]^3
y=Sin[t]^3
find the Cartesian equation for it.\
\>", "Text"]
}, Open  ]],

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Cell["Solution", "Subsection"],

Cell[CellGroupData[{

Cell["Converting the Parametric formula to Cartesian equation", \
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Cell[TextData[{
  "If ",
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  ", ",
  Cell[BoxData[
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  Cell[BoxData[{
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      \(y\^\(2/3\) \[Equal] Sin[t]\^2\)}]],
  "\n(We do not have to worry about the sign in squaring both sides, because \
x==Cos[t]^3 is a definition, not an equation with an unknown.)\nAdding these \
equations results in\n",
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Cell["Expanding the Equation", "Subsubsection",
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Cell[TextData[{
  Cell[BoxData[
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  ", from this we can get several equivalent forms, to be used later. ( (*3*) \
is not strickly equivalent because we applied the non-one-to-one function \
#^(2) to both sides)"
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Cell[TextData[{
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  "\t(*2*)\n",
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  " using (*3*)\nsimplify and apply the function #^3 one more time to get\n",
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Cell[TextData[{
  "here we do it in ",
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Cell[BoxData[
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Cell["\<\
Now we have an equation without fraction powers. Astroid is \
commonly given as
(x^2+y^2-a^2)^3+27 a^2 x^2 y^2==0
which is derived by starting with the parametric formula with a scaling \
factor a. Here we show how to arrive at that form from our equation \
above.\
\>", "Text"],

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Cell[BoxData[
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Cell["\<\
Now, we introduce a scaling factor by replacing x with x/a and y \
with y/a.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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